The differential equation, x(x−x0)(d2y/dx2)+(B1+B2x) (dy/dx)+[ω2x(x−x0) −[2ηω(x−x0)+B3]y=0, arises both in the quantum scattering theory of nonrelativistic electrons from polar molecules and ions, and, in the guise of Teukolsky’s equations, in the theory of radiation processes involving black holes. This article discusses analytic representations of solutions to this equation. Previous results of Hylleraas [E. Hylleraas, Z. Phys. 71, 739 (1931)], Jaffé [G. Jaffé, Z. Phys. 87, 535 (1934)], Baber and Hassé [W. G. Baber and H. R. Hassé, Proc. Cambridge Philos. Soc. 25, 564 (1935)], and Chu and Stratton [L. J. Chu and J. A. Stratton, J. Math. Phys. (Cambridge, Mass.) 20, 3 (1941)] are reviewed, and a rigorous proof is given for the convergence of Stratton’s spherical Bessel function expansion for the ordinary spheroidal wave functions. An integral is derived that relates the eigensolutions of Hylleraas to those of Jaffé. The integral relation is shown to give an integral equation for the scalar field quasinormal modes of black holes, and to lead to irregular second solutions to the equation. New representations of the general solutions are presented as series of Coulomb wave functions and confluent hypergeometric functions. The Coulomb wave‐function expansion may be regarded as a generalization of Stratton’s representation for ordinary spheroidal wave functions, and has been fully implemented and tested on a digital computer. Both solutions given by the new algorithms are analytic in the variable x and the parameters B1, B2, B3, ω, x0, and η, and are uniformly convergent on any interval bounded away from x0. They are the first representations for generalized spheroidal wave functions that allow the direct evaluation of asymptotic magnitude and phase.

1.
W. H.
Press
 and
S. A.
Teukolsky
,
Astrophys. J.
185
,
649
(
1973
).
Google Scholar
Crossref
Search ADS
 
2.
C. W.
Clark
,
Phys. Rev. A
20
,
1875
(
1979
).
Google Scholar
Crossref
Search ADS
 
3.
S. Detweiler, in Sources of Gravitational Radiation, edited by L. Smarr (Cambridge U.P., London, 1979), p. 217.
4.
E. W. Leaver, “A spectral decomposition of the perturbation response of the Schwarzschild geometry,” submitted to Phys. Rev. D.
5.
W. G.
Baber
 and
H. R.
Hassé
,
Proc. Cambridge Philos. Soc.
25
,
564
(
1935
).
Google Scholar
 
6.
W.
Gautschi
,
SIAM Rev.
9
,
24
(
1967
).
Google Scholar
Crossref
Search ADS
 
7.
E.
Hylleraas
,
Z. Phys.
71
,
739
(
1931
).
Google Scholar
Crossref
Search ADS
 
8.
G.
Jaffé
,
Z. Phys.
87
,
535
(
1934
).
Google Scholar
Crossref
Search ADS
 
9.
J. Meixner, F. S. Schäfke, and G. Wolf, “Mathieu Functions and Spheroidal Functions and their Mathematical Foundations (further studies),” in Lecture Notes in Mathematics, Vol. 837 (Springer, Berlin, 1980).
10.
P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw‐Hill, New York, 1953).
11.
E. W.
Leaver
,
Proc. R. Soc. Lond. Ser. A
402
,
285
(
1985
).
Google Scholar
 
12.
H. Eyring, J. Walter, and G. E. Kimball, Quantum Chemistry (Wiley, New York, 1948).
13.
J. A.
Wheeler
,
Phys. Rev.
97
,
511
(
1955
).
Google Scholar
Crossref
Search ADS
 
14.
T.
Regge
 and
J. A.
Wheeler
,
Phys. Rev.
108
,
1063
(
1957
).
Google Scholar
Crossref
Search ADS
 
15.
F. J.
Zerilli
,
Phys. Rev. D
2
,
2141
(
1970
).
Google Scholar
Crossref
Search ADS
 
16.
S.
Chandrasekhar
,
Proc. R. Soc. Lond. Ser. A
343
,
289
(
1975
).
Google Scholar
 
17.
S.
Chandrasekhar
 and
S.
Detweiler
,
Proc. R. Soc. Lond. Ser. A
344
,
441
(
1975
).
Google Scholar
 
18.
S. A.
Teukolsky
,
Phys. Rev. Lett.
29
,
1114
(
1972
).
Google Scholar
Crossref
Search ADS
 
19.
R. H.
Price
,
Phys. Rev. D
5
,
2419
(
1972
).
Google Scholar
Crossref
Search ADS
 
20.
C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (Freeman, San Francisco, 1973), exercise 32.10.
21.
S. Chandrasekhar, The Mathematical Theory of Black Holes (Clarendon, Oxford, 1983).
22.
A. N. Lowan, in Handbook of Mathematical Functions, edited by M. Abramowitz and I. Stegun (National Bureau of Standards, Washington, 1964), Chap. 21. There are other definitions for the oblate coordinates λ and μ.
23.
E. D.
Fackerell
 and
R. G.
Crossman
,
J. Math. Phys.
18
,
1849
(
1977
).
Google Scholar
Crossref
Search ADS
 
24.
R. A.
Breuer
,
M. P.
Ryan
, and
S.
Waller
,
Proc. R. Soc. Lond. Ser. A
358
,
71
(
1977
).
Google Scholar
 
25.
C.
Hunter
 and
B.
Guerrieri
,
Stud. Appl. Math.
66
,
217
(
1982
).
Google Scholar
Crossref
Search ADS
 
26.
V.
Ferrari
 and
B.
Mashhoon
,
Phys. Rev. D
30
,
295
(
1984
).
Google Scholar
Crossref
Search ADS
 
27.
A. H.
Wilson
,
Proc. R. Soc. Lond. Ser. A
118
,
617
(
1928
).
Google Scholar
 
28.
L. J. Slater, in Handbook of Mathematical Functions, edited by M. Abramowitz and I. Stegun (National Bureau of Standards, Washington, 1964), Chap. 13.
29.
I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965).
30.
S.
Chandrasekhar
,
Proc. R. Soc. Lond. Ser. A
392
,
1
(
1984
).
Google Scholar
 
31.
A. L. Rabenstein, Introduction to Ordinary Differential Equations (Academic, New York, 1966).
32.
E. L. Ince, Ordinary Differential Equations (Dover, New York, 1926), p. 201.
33.
I. S. Gradshteyn and I. M. Ryzhik, Ref. 29, Eq. 7.414.8.
34.
J. A.
Stratton
,
Proc. Natl. Acad. Sci. U.S.A.
21
,
51
,
316
(
1935
). We use the term “Stratton solution” somewhat loosely, as a very important generalization was made in a subsequent paper coauthored by L. J. Chu (see Ref. 38).
Google Scholar
Crossref
Search ADS
PubMed
 
Contributions were also made by Morse [
P. M.
Morse
,
Proc. Natl. Acad. Sci. U.S.A.
21
,
56
(
1935
): also see Ref. 10].
Google Scholar
Crossref
Search ADS
PubMed
 
35.
P. M. Morse and H. Feshbach, Ref. 10, Eq. 11.3.87.
36.
P. M. Morse and H. Feshbach, Ref. 10, Eq. 11.3.39.
37.
P. M. Morse and H. Feshbach, Ref. 10, p. 1503. There are brackets missing from the expression for βn in some editions of the book.
38.
The generalized expansions (86) and (87) were first given by Chu and Stratton [
L. J.
Chu
 and
J. A.
Stratton
,
J. Math. Phys. (Cambridge, Mass.)
20
,
259
(
1941
)],
Google Scholar
 
and reprinted in Spheroidal Wave Functions [J. A. Stratton, P. M. Morse, L. J. Chu, J. D. C. Little, and F. J. Corbató (Wiley, New York, 1956)].
Our present equations (89) and (90) are completely equivalent to Eqs. (95) and (96) of Chu and Stratton, the important difference being that in the present study we regard the separation constant Alm as being fixed, and use these equations to define the characteristic value of the phase parameter ν.
39.
P. M. Morse and H. Feshbach, Ref. 10, p. 1507.
40.
B. P.
Sinha
 and
R. H.
Macphie
,
J. Math. Phys.
16
,
2378
(
1975
).
Google Scholar
Crossref
Search ADS
 
41.
L.
Jen
 and
C.
Hu
,
IEEE Trans. Antennas Propag.
31
,
382
(
1983
).
Google Scholar
 
42.
R.
Penrose
,
Nuovo Cimento
1
(special number),
252
(
1969
).
Google Scholar
 
43.
S. A.
Teukolsky
 and
W. H.
Press
,
Astrophys. J.
193
,
443
(
1974
).
Google Scholar
Crossref
Search ADS
 
44.
S.
Detweiler
,
Astrophys. J.
239
,
292
(
1980
).
Google Scholar
Crossref
Search ADS
 
45.
A. R.
Barnett
,
D. H.
Feng
,
J. W.
Steed
, and
L. J. B.
Goldfarb
,
Comput. Phys. Commun.
8
,
377
(
1974
).
Google Scholar
Crossref
Search ADS
 
46.
E. W. Leaver, “GSWFN—Generalized spheroidal wave‐functions for complex arguments,” to be submitted to Trans. Math. Software.
47.
A. Messiah, Quantum Mechanics (North‐Holland, Amsterdam, 1958).
48.
E. C. Titchmarsh, Eigenfunction Expansions Associated with Second Order Differential Equations (Oxford U.P., New York, 1962).
49.
E. L. Ince, Ref. 32, Chap. 5.
This content is only available via PDF.
© 1986 American Institute of Physics.
1986
American Institute of Physics
You do not currently have access to this content.